2320 IEEE TRANSACTIONS ON INSTRUMENTATION AND …2320 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 55, NO. 6, DECEMBER 2006 Hilbert–Huang Transform-Based Vibration

2320 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 55, NO. 6, DECEMBER 2006

Hilbert–Huang Transform-Based Vibration SignalAnalysis for Machine Health Monitoring

Ruqiang Yan, Student Member, IEEE, and Robert X. Gao, Senior Member, IEEE

Abstract—This paper presents a signal analysis technique formachine health monitoring based on the Hilbert-Huang Trans-form (HHT). The HHT represents a time-dependent series ina two-dimensional (2-D) time-frequency domain by extractinginstantaneous frequency components within the signal throughan Empirical Mode Decomposition (EMD) process. The analyt-ical background of the HHT is introduced, based on a syntheticanalytic signal, and its effectiveness is experimentally evaluatedusing vibration signals measured on a test bearing. The resultsdemonstrate that HHT is suited for capturing transient events indynamic systems such as the propagation of structural defects in arolling bearing, thus providing a viable signal processing tool formachine health monitoring.

Index Terms—Empirical mode decomposition, Hilbert–Huangtransform, intrinsic mode function, machine health monitoring,transient signal analysis.

I. INTRODUCTION

AS a critical component in automated machine systems, ef-fective machine health monitoring has attracted increasing

attention from the research community over the past decade[1]–[4]. The goal of health monitoring is to identify potentialfailure causes at the early stage such that timely adjustmentand maintenance actions can be taken to reduce severe machinedamage and costly machine downtime. Of the various moni-toring techniques commonly employed, vibration measurementremains an effective approach. This is due to the fact that struc-tural failures in a machine system cause changes to the dynamiccharacteristics of the machine, which is reflected in its vibra-tion signals. By means of appropriate signal decomposition andrepresentation, features hidden in the vibration signals can beextracted, and assessment of the machine health status can bemade.

Vibration signals encountered in rotary machine systems,such as machine tools or electric motors, can be broadly clas-sified as being either stationary or nonstationary. Stationarysignals are characterized by time-invariant statistic properties,e.g., the mean values and/or autocorrelation function [5]. Suchsignals can be adequately analyzed using well-known spectraltechniques based on the Fourier Transform. An example is

Manuscript received June 15, 2005; revised September 15, 2006. Thiswork was supported by the National Science Foundation under ContractDMI-0218161. The experimental setup was supported by the SKF and TimkenCompanies.

The authors are with the Department of Mechanical and Industrial En-gineering, University of Massachusetts, Amherst, MA 01003 USA (e-mail:[email protected]; [email protected]).

Color versions of Figs. 1–3, 5–12, and 14–19 are available online at http://ieeexplore.ieee.org

Digital Object Identifier 10.1109/TIM.2006.887042

Fig. 1. Periodic vibration caused by bearing misalignment. (a) Bearing mis-alignment. (b) Misalignment-related vibration and its spectrum.

given in Fig. 1, where bearing misalignment caused by theinner and outer raceways falling out of the same plane resultedin periodic vibrations, which can be identified in its spectrum(see Fig. 1(b)). In contrast, nonstationary signals are “transient”in nature, with duration generally shorter than the observationinterval. Such signals can be generated by the sudden breakageof a drilling bit, flaking of the raceway of a rolling bearing,or a growing crack inside a work piece. Effective detection ofnonstationary signals is of great importance, as they are pre-cursors of potential machine failures. However, their temporary

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YAN AND GAO: HILBERT–HUANG TRANSFORM-BASED VIBRATION SIGNAL ANALYSIS FOR MACHINE HEALTH MONITORING 2321

nature makes the assumption of signal stationarity as requiredby Fourier transform invalid, thus reducing its effectiveness.

In recent years, time-frequency analysis techniques, suchas the Short-Time Fourier Transform (STFT) [6] and wavelettransform [7]–[9], have been investigated for their applicabilityin feature extraction from nonstationary, transient signals. Suc-cessful application of these techniques requires understandingof their respective limitations. For example, selection of asuitable window size is required when applying the STFTto match with the specific frequency content of the signal,which is generally not known a priori. When applying thewavelet transform, the type of the basic wavelet functionemployed directly affects the effectiveness in identifyingtransient elements hidden within the dynamic signal [10]. Incomparison, the Hilbert–Huang transform (HHT) is based onthe instantaneous frequencies resulting from the intrinsic modefunctions of the signal being analyzed [11]–[13]; thus, it is notconstrained by the uncertainty limitations with respect to thetime and frequency resolutions to which other time-frequencytechniques are subject. In recent years, HHT has been appliedto identification of damage time instant and location in civiland mechanical structures [14]–[17]. Using HHT, the physicalmass, damping coefficient, and stiffness matrices of a multipledegree of freedom (MDOF) linear system could be identified[18], [19]. In the area of biomedical engineering, studies on res-piratory sinus arrhythmia from analyzing heartbeat time series[20] and heart rate variability [21] have further demonstratedthe effectiveness of HHT in transient signal processing.

This paper investigates the utility of HHT as a tool for vibra-tion signal analysis, with machine health monitoring as a targetapplication area. After introducing the theoretical background,a synthetic signal is analytically formulated and used as a testsignal for evaluating the performance of HHT. The simulation issubsequently evaluated through experimental studies performedon a bearing test bed. The results demonstrate the effectivenessof HHT for signal decomposition and feature extraction in ma-chine health monitoring applications.

II. THEORETICAL BACKGROUND

The HHT represents the signal being analyzed in the time-fre-quency domain by combining the empirical mode decomposi-tion (EMD) with the Hilbert transform [11]. In contrast to theFourier spectral analysis by which a series of sine and cosinefunctions of constant amplitudes are used to represent each con-stituent frequency components in the signal, the HHT techniqueis based on the instantaneous frequency calculation that resultsfrom the Hilbert transform of the signal. Generally, the Hilberttransform for any signal is defined as

(1)

where denotes the Hilbert transform operation. Theoreti-cally, any analytic signal can be expressed by the sum of itsreal part and imaginary part , with the latter being theHilbert transform of the real part. This results in

(2)

Fig. 2. Instantaneous frequency of the vibration signal caused by bearing mis-alignment.

Fig. 3. Vibration signal measured from a ball bearing shown in the time andfrequency domains.

Equation (2) can be rewritten in a polar coordinate system as

(3)

where

(4)

represents the instantaneous amplitude and phase of the analyticsignal, respectively. From the instantaneous phase , the in-stantaneous frequency of the signal can be derived as

(5)


Fig. 4. Flow chart of the EMD process for signal decomposition.

Accordingly, the real part of the signal can be written interms of the amplitude and instantaneous frequency as a time-dependent function

(6)

where the symbol denotes the real part of the analyticsignal .

To ensure that the instantaneous frequency obtained from thederivative operation in (5) is physically meaningful, the instan-taneous phase must be a single-valued function over time,i.e., a mono-component function. Since the instantaneous phase

is constructed from and its Hilbert transform, as shownin (4), it follows that must also be a mono-component func-tion. The periodic vibration signal shown in Fig. 1 satisfies thisrequirement; thus, its instantaneous frequency can be calculatedby using (3)–(5). The instantaneous frequency of 20 Hz is shownin the time-frequency representation (see Fig. 2) as a straightline across the data acquisition period, which is consistent withthe time scale in Fig. 1.

To effectively construct frequency spectrum of a vibrationsignal that contains multiple-frequency components, the signalneeds to be first decomposed into mono-component functions,by means of EMD [11]. A vibration signal measured from aball bearing is shown in Fig. 3. Its spectrum reveals two compo-nents at 20 and 41 Hz, respectively, which are associated withthe bearing misalignment and ball pass frequency in the outerraceway (caused by the periodic passing of the rolling balls overa fixed reference position).

Decomposition of such a signal is based on the following ob-servations.

1) The signal has at least two extrema, i.e., one maximum andone minimum.

Fig. 5. Intrinsic mode functions and the residue of a bearing vibration signal.

Fig. 6. Reconstructed bearing vibration signal and the amplitude error. (a)Residue+IMF3. (b) Residue+IMF3+IMF2. (c) Residue+IMF3+IMF2+IMF1.(d) Amplitude error.

2) The characteristic time scale is clearly defined by the timelapse between successive alternations of local maxima andminima of the signal.

3) If the signal has no extrema but contains inflection points,then it can be differentiated one or more times to reveal theextrema.

The EMD technique decomposes the signal into a numberof Intrinsic Mode Functions (IMFs), each of which a mono-component function. Then, the Hilbert transform is applied tocalculate the instantaneous frequencies of the original signal.

The procedure for extracting the IMFs from a signal isillustrated in Fig. 4. After identifying all the local maxima andminima of the signal, the upper and lower envelopes are gener-ated through curve fitting. Research has shown that many com-plex curve fitting functions have only resulted in marginal im-provement while increasing the computational load significantly


Fig. 7. HHT of the vibration signal.

TABLE IPARAMETERS FOR CONSTRUCTING THE SYNTHETIC TEST SIGNAL

[11]. Therefore, the cubic spline function was employed in thepresented study. The mean values of the upper and lower en-velopes of the signal are calculated as

(7)

where and are the upper and lower envelopesof the signal, respectively. Accordingly, the difference betweenthe signal and the envelopes of the signal , which isdenoted as , is given by

(8)

Due to the approximation nature of the curve fitting method,has to be further processed (by treating as the

signal itself and repeating the process continually) until it satis-fies the following two conditions.

1) The number of extrema and the number of zero-crossingsare either equal to each other or differ by at most one.

2) At any point, the mean value between the envelope de-fined by local maxima and the envelope defined by the localminima is zero.

Through the iteration process (for a total of times), the dif-ference between the signal and the mean envelope values, whichis denoted as , is obtained as

(9)

Fig. 8. Synthetic test signal and its spectrum. (a) Time (in milliseconds).(b) Frequency (in Hertz).

Fig. 9. Intrinsic mode functions of the synthetic test signal extracted by EMD.

where is the mean envelope value after the th iteration,and is the difference between the signal and the meanenvelope values at the th iteration. The functionis then defined as the first IMF component and expressed as

(10)

After separating from the original signal , the residueis obtained as

(11)


Fig. 10. HHT of the synthetic test signal.

Subsequently, the residue can be treated as the new signal,and the above-illustrated iteration process is repeated to extractthe rest of the IMFs inherent to the signal as

... (12)

The signal decomposition process is terminated when be-comes a monotonic function, from which no further IMFs canbe extracted. By substituting (12) into (11), the signal isdecomposed into a number of intrinsic mode functions that arethe constituent components of the signal. As a result, the signal

can be expressed as

(13)

where represents the th intrinsic mode function, andis the residue of the signal decomposition. Equation (13) pro-vides a complete description of the empirical mode decomposi-tion process [11], which can be evaluated by checking the am-plitude error between the reconstructed and the original signal.As an example, Fig. 5 illustrates both the IMFs and the residueof the multicomponent bearing vibration signal that is initiallyshown in Fig. 3. The reconstructed vibration signal is shown inFig. 6, and the amplitude error between the original and recon-structed signals is calculated and illustrated in Fig. 6(d). Giventhat the maximum amplitude error is in the order of (V),the EMD signal decomposition process can be considered to becomplete.

Based on (1)–(5) and (13), (6) can be modified as

(14)

Fig. 11. Short-time Fourier transform of the synthetic signal. (a) Window size:1.6 ms. (b) Window size: 25.6 ms. (c) Window size: 6.4 ms.

in which and. The term in (13) is not

included in (14) as it is a monotonic function and does notcontribute to the frequency content of the signal.


Fig. 12. Wavelet transform of the synthetic signal. (a) Basic wavelet function:Db2. (b) Basic wavelet function: Meyer. (c) Basic wavelet function: Morlet.

Comparing (14) with the Fourier-based representation of asignal given by

(15)

Fig. 13. Experimental test bed.

where both and are constant, it becomes evident thatthe EMD process enables flexible representation of a dynamicsignal by revealing its time-dependent amplitude and the char-acteristic frequency components at various time instances. Thesignal is thus represented by a time-frequency distribution. Theunderlying HHT of the signal is mathematically defined as

HHT HHT (16)

where HHT represents the time-frequency distributionobtained from the th IMF of the signal. The symbol denotes“by definition,” and combines the amplitude andinstantaneous frequency of the signal together.

In Fig. 7, the HHT result of the initial vibration signal (givenin Fig. 3) is illustrated. The two major frequency componentsat 20 and 41 Hz were identified as constants throughout the ob-servation period of 200 ms. Comparing to the Fourier spectrum,the time-frequency diagram has the advantage in revealing thetime-dependency of the frequency components of interest. Suchability is particularly useful when analyzing transient signalswhose constituent frequencies may change over time, e.g., whenthe health condition of a machine degrades due to defect growthand propagation.

In the above analysis, the test signal decomposed using theHHT process was experimentally measured from a realisticball bearing. To generalize the concept of HHT-based vibrationsignal decomposition as a viable and effective signal processingtool for machine health monitoring, a quantifiable signal isneeded to provide a computational reference base. For this pur-pose, a synthetic test signal was constructed, which consistedof defect-induced transient components with time-varying fre-quencies [10]. Amplitude variation of the transient componentsprovides an indication of the bearing defect degradation. Giventhat defect-induced vibrations are generated every time whenthe rolling elements and the defect interact, the transient com-


Fig. 14. IMFs of vibration signals from two roller bearings. (Left) Healthy bearing. (Right) Defective bearing.

ponents were formulated as a series of exponentially decayingoscillations [10], [22], given by:

(17)

with being the number of transient elements, and, and corresponding to the amplitude, attenuation factor,

time-delay, initial phase, and frequency of the th transientcomponent, respectively. The specific values of these parame-ters used for constructing the synthetic test signal are listed inTable I. In Fig. 8, the synthetic test signal is shown.

As seen in Fig. 8, a total of four groups of impulsive signaltrains were created to simulate a transient vibration signal. Eachof the pulse groups contains transient elements of two differentcenter frequencies. For example, in the first two pulse groups,the center frequencies of the transient elements are equal to 1500and 700 Hz, respectively. Similarly, the second two pulse groupswere formulated using center frequencies at 1600 and 600 Hz.The four groups are separated from each other by a 12–ms timeinterval. Within each group, the two transient elements are timeoverlapped.

As shown in Fig. 8, the Fourier-based spectrum, althoughit reveals the four center frequencies contained in the signal,does not identify any frequency changes of the transient com-ponents. In comparison, the time-frequency decomposition ofthe signal based on HHT has revealed that between the time in-terval of 20–30 ms, which is the frequency composition of the

signal changed from containing a 1500-Hz and a 700-Hz com-ponent to containing a 1600-Hz and a 600-Hz component. Sucha result demonstrates the effectiveness of HHT in analyzingtransient vibration components. The IMFs extracted during theEMD process are shown in Fig. 9, whereas in Fig. 10, the time-frequency decomposition of the various frequency componentsembedded in the synthetic test signal using HHT is illustrated.

To enable quantitative comparison between HHT and othertime-frequency analysis techniques, the same signal was ana-lyzed using the STFT and wavelet transform, as illustrated inFigs. 11 and 12 respectively. It can be seen that only when ap-propriate window size (i.e., 6.4 ms) and basic wavelet function(i.e., Morlet wavelet) were chosen could the transient compo-nents existing in the signal be successfully identified and dif-ferentiated. Such results reveal the constraints associated withthese alternative time-frequency techniques, and comparativelyindicate that HHT provides a viable approach to transient signalanalysis.

III. EXPERIMENTAL EVALUATION

To experimentally evaluate the effectiveness of HHT fornonstationary vibration signal analysis, systematic tests wereconducted on a custom-built bearing test bed (see Fig. 13). Thetest bed consisted of a DC motor (Lesson Cl6D34FT18), twosupporting pillow blocks (SKF 209-112), a test bearing withsupporting housing, a hydraulic cylinder (Miller 4Z644B), ahydraulic pump (Enerpac P392), and an optical encoder for


Fig. 15. HHT of vibration signals from two roller bearings (Left) Healthy bearing signal. (Right) Defective bearing signal.

bearing speed measurement. The hydraulic cylinder appliesvariable load to the bearing in the radial direction. An ac-celerometer (bandwidth 1 Hz–30 kHz) measured vibrationsfrom two test bearings (ball and roller bearing), which con-tained seeded defects on their raceways.

A. Defect Identification

The ability of the HHT for detecting bearing defect was firstinvestigated on a roller bearing with a seeded defect (0.27 mmgroove across its outer raceway). Shown in Fig. 14 is a compar-ison of extracted IMFs between vibration signals from a refer-ence “healthy” bearing [without structural defect: Fig. 14 (left)]and a defective bearing [Fig. 14 (right)]. The corresponding re-sults of the HHT analysis are illustrated in Fig. 15. For the de-fective bearing, transient vibrations caused by the rollers-defectinteractions are clearly seen in the frequency range of 1–5 kHzand 8–12 kHz [see Fig. 15 (right). In addition, these transientshave shown a repetitive pattern with a 6-ms interval, whichcorresponds to a 169-Hz repetitive BPFO frequency, resultingfrom the structural defect on the outer raceway. The healthybearing, in comparison, showed no high-frequency componentsor repetitive signal patterns [see Fig. 15 (left), since no defectwas present.

B. Defect Degradation Diagnosis

A run-to-failure experiment was conducted on a deep grooveball bearing of 52 mm outer diameter, in which a 0.27-mm-widegroove was cut across its outer raceway. Upon reaching ap-proximately 2.7 million revolutions, the defect had propagatedthroughout the entire raceway and rendered the bearing prac-tically nonfunctional. Vibration signals were taken during thecourse of the experiment, at an interval of every 7 min, to mon-itor the defect propagation process. Fig. 16 illustrates two datasegments, covering the transitions of bearing test from phase Ito II and from phase II to III.

The Fourier spectrum of the two data segments, shown inFig. 17, illustrates averaged frequency information over the en-tire data sampling period, without providing specific informa-tion on defect propagation. In comparison, Fig. 18 illustrates

Fig. 16. Amplitude as a function of the bearing revolution.

the IMFs of the two data segments. The corresponding HHTidentified frequency changes at the time instance of 45 ms [seeFig. 19 (left)] during the first data segment and at 55 ms duringthe second data segment [see Fig. 19 (right)]. Such frequencychanges reflect degradation of the bearing health condition asthe defect propagated through the bearing raceway. Physically,impacts generated by the rolling ball-defect interactions exciteintrinsic modes of the bearing system, giving rise to a train oftransient vibrations at the mode-related resonant frequencies. Asthe defect size increase, different intrinsic modes would be ex-cited, resulting in the change of frequency components in thetransient vibrations. In addition, these transients have shown arepetitive pattern with a 15-ms interval, which corresponds toa 67-Hz frequency component that is related to a bearing mis-alignment. Thus, the HHT has shown to provide an effective toolfor bearing health diagnosis.

IV. CONCLUSION

As a time-frequency signal decomposition technique, theHilbert-Huang transform provides an effective tool for ana-lyzing transient vibration signal. Numerical and experimentalstudies on a custom-built bearing test bed have shown that


Fig. 17. Fourier spectrum of the two data segments. (Left) Segment 1. (Right) Segment 2.

Fig. 18. IMFs of the two data segments (Left) Segment 1. (Right) Segment 2.

Fig. 19. HHT of the two data segments. (Left) Segment 1. (Right) Segment 2.


the deterioration of a test bearing can be effectively detectedthrough the time-dependent amplitudes and instantaneous fre-quencies resulting from the HHT. Such a technique can be fur-ther applied to the health monitoring of other types of dynamicsystems, such as electrical drives.

The computational load of the HHT technique is determinedby the EMD of the signal being analyzed, as it is an iterationprocess. To obtain a general idea on the computational efficiencyof the HHT for vibration signal analysis, experiments were con-ducted on a laptop computer with a 2.0-GHz CPU and 1.0-GBmemory. For the various signals investigated in the presentedstudy, the computational time taken varied from 0.3 to 1.8 sec,which confirms the general applicability of the HHT techniquefor online machine health monitoring.

Research is being continued to systematically investigate thesuitability and constraints of the HHT for nonstationary signalanalysis, using vibration signals from different types of bear-ings. Furthermore, integration of the EMD process with en-veloping spectrum analysis is also being investigated, with thegoal of realizing an adaptive signal demodulation approach toaccount for varying machine conditions in real-world applica-tions.

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Ruqiang Yan (S’04) received the B.S. and M.S. de-grees in mechanical engineering from the Universityof Science and Technology of China, Hefei, China,in 1997 and 2002, respectively. He is currentlyworking toward the Ph.D. degree at the Departmentof Mechanical and Industrial Engineering, Univer-sity of Massachusetts, Amherst.

He is a Research Assistant at the Department ofMechanical and Industrial Engineering, Universityof Massachusetts. His research interests includenonlinear time-series analysis, multidomain signal

processing, machine condition monitoring and health diagnosis, sensors, andsensor networks.

Robert X. Gao (M’91-SM’00) received the Ph.D.and M.S. degrees from the Technical UniversityBerlin (TU Berlin), Berlin, Germany, in 1991, and1985, respectively.

Presently, he is a Professor with the Departmentof Mechanical and Industrial Engineering, Univer-sity of Massachusetts, Amherst. His research andteaching interests include physics-based sensingmethodology, self-diagnostic and energy-efficientsensors and sensor networks, mechatronic systemsdesign, medical instrumentation, and wavelet trans-forms for machine health monitoring, diagnosis, and

prognosis.Dr. Gao received the National Science Foundation CAREER Award in

1996, is a faculty advisor and co-recipient of the inaugural Best Student PaperAward from the SPIE’s International Symposium on Smart Structures andMaterials in 1997, and received the University of Massachusetts OutstandingEngineering Junior Faculty Award in 1999. He is a Co-Editor of the bookCondition Monitoring and Control for Intelligent Manufacturing (SpringerVerlag: 2006). He was the Guest Editor for a Special Section on Built-In-Testfor the IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT,published in June 2005, and the Guest editor for a Special Section on Sensorsfor the Journal of Dynamic Systems, Measurement, and Control, publishedby the American Society of Mechanical Engineers (ASME) in June 2004.Currently, he serves as an Associate Editor for the IEEE TRANSACTIONS ONINSTRUMENTATION AND MEASUREMENT and an Associate Editor for theASME Journal of Dynamic Systems, Measurement, and Control. He is also onthe Editorial Board of the International Journal of Manufacturing Research.He is a co-chair of the Technical Committee on Built-in-Test and Self-Test ofthe IEEE Instrumentation and Measurement Society.

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2320 IEEE TRANSACTIONS ON INSTRUMENTATION AND …2320 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 55, NO. 6, DECEMBER 2006 Hilbert–Huang Transform-Based Vibration